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Properties

Label 465.2.c.b
Level $465$
Weight $2$
Character orbit 465.c
Analytic conductor $3.713$
Analytic rank $0$
Dimension $22$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [465,2,Mod(94,465)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(465, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("465.94");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.c (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.71304369399\)
Analytic rank: \(0\)
Dimension: \(22\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 30 q^{4} + 6 q^{6} - 22 q^{9} - 4 q^{10} + 28 q^{14} + 2 q^{15} + 54 q^{16} - 24 q^{19} - 14 q^{20} + 16 q^{21} - 18 q^{24} - 10 q^{25} - 12 q^{26} + 16 q^{29} + 4 q^{30} + 22 q^{31} - 4 q^{34} - 4 q^{35}+ \cdots + 42 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
94.1 − 2.73720i 1.00000i −5.49226 −0.841130 − 2.07184i 2.73720 2.97118i 9.55900i −1.00000 −5.67102 + 2.30234i
94.2 − 2.61420i − 1.00000i −4.83402 −0.308027 + 2.21475i −2.61420 4.28641i 7.40869i −1.00000 5.78979 + 0.805243i
94.3 − 2.60996i 1.00000i −4.81187 1.75621 + 1.38410i 2.60996 − 1.35174i 7.33885i −1.00000 3.61243 − 4.58364i
94.4 − 2.15913i 1.00000i −2.66183 −1.85009 + 1.25586i 2.15913 − 2.73322i 1.42896i −1.00000 2.71156 + 3.99457i
94.5 − 2.11614i − 1.00000i −2.47806 1.31056 − 1.81175i −2.11614 1.31151i 1.01163i −1.00000 −3.83391 − 2.77334i
94.6 − 1.65822i − 1.00000i −0.749697 −0.117620 − 2.23297i −1.65822 0.225788i − 2.07328i −1.00000 −3.70276 + 0.195040i
94.7 − 1.34956i 1.00000i 0.178679 1.58199 − 1.58029i 1.34956 3.66966i − 2.94027i −1.00000 −2.13270 − 2.13499i
94.8 − 1.31655i 1.00000i 0.266704 2.22783 − 0.191783i 1.31655 − 4.78991i − 2.98422i −1.00000 −0.252491 − 2.93304i
94.9 − 0.523109i − 1.00000i 1.72636 0.328261 + 2.21184i −0.523109 3.11438i − 1.94929i −1.00000 1.15703 − 0.171716i
94.10 − 0.364910i − 1.00000i 1.86684 −2.16572 + 0.556482i −0.364910 0.715024i − 1.41105i −1.00000 0.203066 + 0.790293i
94.11 − 0.104188i 1.00000i 1.98914 −1.92227 + 1.14231i 0.104188 3.88714i − 0.415622i −1.00000 0.119015 + 0.200278i
94.12 0.104188i − 1.00000i 1.98914 −1.92227 − 1.14231i 0.104188 − 3.88714i 0.415622i −1.00000 0.119015 − 0.200278i
94.13 0.364910i 1.00000i 1.86684 −2.16572 − 0.556482i −0.364910 − 0.715024i 1.41105i −1.00000 0.203066 − 0.790293i
94.14 0.523109i 1.00000i 1.72636 0.328261 − 2.21184i −0.523109 − 3.11438i 1.94929i −1.00000 1.15703 + 0.171716i
94.15 1.31655i − 1.00000i 0.266704 2.22783 + 0.191783i 1.31655 4.78991i 2.98422i −1.00000 −0.252491 + 2.93304i
94.16 1.34956i − 1.00000i 0.178679 1.58199 + 1.58029i 1.34956 − 3.66966i 2.94027i −1.00000 −2.13270 + 2.13499i
94.17 1.65822i 1.00000i −0.749697 −0.117620 + 2.23297i −1.65822 − 0.225788i 2.07328i −1.00000 −3.70276 − 0.195040i
94.18 2.11614i 1.00000i −2.47806 1.31056 + 1.81175i −2.11614 − 1.31151i − 1.01163i −1.00000 −3.83391 + 2.77334i
94.19 2.15913i − 1.00000i −2.66183 −1.85009 − 1.25586i 2.15913 2.73322i − 1.42896i −1.00000 2.71156 − 3.99457i
94.20 2.60996i − 1.00000i −4.81187 1.75621 − 1.38410i 2.60996 1.35174i − 7.33885i −1.00000 3.61243 + 4.58364i
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 94.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 465.2.c.b ✓ 22
3.b odd 2 1 1395.2.c.h 22
5.b even 2 1 inner 465.2.c.b ✓ 22
5.c odd 4 1 2325.2.a.bc 11
5.c odd 4 1 2325.2.a.bd 11
15.d odd 2 1 1395.2.c.h 22
15.e even 4 1 6975.2.a.ci 11
15.e even 4 1 6975.2.a.cj 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
465.2.c.b ✓ 22 1.a even 1 1 trivial
465.2.c.b ✓ 22 5.b even 2 1 inner
1395.2.c.h 22 3.b odd 2 1
1395.2.c.h 22 15.d odd 2 1
2325.2.a.bc 11 5.c odd 4 1
2325.2.a.bd 11 5.c odd 4 1
6975.2.a.ci 11 15.e even 4 1
6975.2.a.cj 11 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{22} + 37 T_{2}^{20} + 582 T_{2}^{18} + 5076 T_{2}^{16} + 26847 T_{2}^{14} + 88469 T_{2}^{12} + \cdots + 25 \) acting on \(S_{2}^{\mathrm{new}}(465, [\chi])\). Copy content Toggle raw display